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1.2 Chemical Potential in an Ideal Gas Mixture An ideal gas mixture is a gas mixture having the following properties: 1) The equation of state PV=n tot RT obeyed for all T, P & compositions. (n tot = total no. moles of gas). 2) If the mixture is separated from pure gas i by a thermally conducting rigid membrane permeable to gas i only, at equilibrium the partial pressure of gas i in the mixture is equal to the pure-gas-i system. 4 At equilibrium, P* i = P i Mole fraction of i(n i /n tot )

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1.2 Chemical Potential in an Ideal Gas Mixture Let μ i – the chemical potential of gas i in the mixture Let μ* i – the chemical potential of the pure gas in equilibrium with the mixture through the membrane. The condition for phase equilibrium: The mixture is at T & P, has mole fractions x 1, x 2,….x i The pure gas i is at temp, T & pressure, P* i. P* i at equilibrium equals to the partial pressure of i, P i in the mixture: Phase equilibrium condition becomes: gas in the mixture pure gas (ideal gas mixture) 5 At equilibrium, P* i = P i

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Example 1 A mixture of mmol of H2S & 5.48mmol of CH4 was placed in an empty container along with a Pt catalyst & the equilibrium was established at C & 762 torr. The reaction mixture was removed from the catalyst & rapidly cooled to room temperature, where the rates of the forward & reverse reactions are negligible. Analysis of the equilibrium mixture found mmol of CS2. Find & for the reaction at C. 11 1bar =750torr

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3. Temperature Dependence of the Equilibrium Constant The ideal-gas equilibrium constant (Kp 0 ) is a function of temperature only. Differentiation with respect to T: From 14

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3. Temperature Dependence of the Equilibrium Constant Since, This is the Vant Hoff equation. The greater the | ΔH 0 |, the faster changes with temperature. Integration: Neglect the temperature dependence of ΔH 0, 15

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Example 2 Find at 600K for the reaction by using the approximation that ΔH 0 is independent of T; Note: 16 Substance kJ/mol NO2 (g) N2O4 (g)

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Answer (Example 2) 17 If ΔH 0 is independent of T, then the vant Hoff equation gives From

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3. Temperature Dependence of the Equilibrium Constant Since, the vant Hoff equation can be written as: The slope of a graph of ln K p 0 vs 1/T at a particular temperature equals –ΔH 0 /R at that temperature. If ΔH 0 is essentially constant over the temperature range, the graph of lnK p 0 vs 1/T is a straight line. The graph is useful to find ΔH 0 if Δ f H 0 of all the species are not known. 18

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Example 3 Use the plot ln K p 0 vs 1/T for for temperature in the range of 300 to 500K Estimate the ΔH Plot of lnK p 0 vs 1/T

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4. Ideal-Gas Equilibrium Calculations Thermodynamics enables us to find the K p 0 for a reaction without making any measurements on an equilibrium mixture. K p 0 - obvious value in finding the maximum yield of product in a chemical reaction. If ΔG T 0 is highly positive for a reaction, this reaction will not be useful for producing the desired product. If ΔG T 0 is negative or only slightly positive, the reaction may be useful. A reaction with a negative ΔG T 0 is found to proceed extremely slow - + catalyst 21

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4. Ideal-Gas Equilibrium Calculations The equilibrium composition of an ideal gas reaction mixture is a function of : T and P (or T and V). the initial composition (mole numbers) n 1,0,n 2,0 ….. Of the mixture. The equilibrium composition is related to the initial composition by the equilibrium extent of reaction (ξ eq ). Our aim is to find ξ eq. 22

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4. Ideal-Gas Equilibrium Calculations 3) Use the stoichiometry of the reaction to express the equilibrium mole numbers (n i ) in terms of the initial mole number (n i,0 ) & the equilibrium extent of reaction (ξ eq ), according to n i =n 0 +ν i ξ eq. 4) (a) If the reaction is run at fixed T & P, use (if P is known) & the expression for n i from n i =n 0 +ν i ξ eq to express each equilibrium partial pressure P i in term of ξ eq. (b) If the reaction is run at fixed T & V, use P i =n i RT/V (if V is known) to express each Pi in terms of ξ eq 24